 2.6.1: Find the variation constant and an equation of variation for the gi...
 2.6.2: Find the variation constant and an equation of variation for the gi...
 2.6.3: Find the variation constant and an equation of variation for the gi...
 2.6.4: Find the variay varies inversely as x, and y = 12 when x = 5tion co...
 2.6.5: Find the variation constant and an equation of variation for the gi...
 2.6.6: Find the variation constant and an equation of variation for the gi...
 2.6.7: Find the variation constant and an equation of variation for the gi...
 2.6.8: Find the variation constant and an equation of variation for the gi...
 2.6.9: Find the variation constant and an equation of variation for the gi...
 2.6.10: Find the variation constant and an equation of variation for the gi...
 2.6.11: Find the variation constant and an equation of variation for the gi...
 2.6.12: Find the variation constant and an equation of variation for the gi...
 2.6.13: Sales Tax. The amount of sales tax paid on a product is directly pr...
 2.6.14: Childs Allowance. The Gemmers decide to give their children a weekl...
 2.6.15: Beam Weight. The weight W that a horizontal beam can support varies...
 2.6.16: Rate of Travel. The time t required to drive a fixed distance varie...
 2.6.17: Fat Intake. The maximum number of grams of fat that should be in a ...
 2.6.18: U.S. House of Representatives. The number of representatives N that...
 2.6.19: Work Rate. The time T required to do a job varies inversely as the ...
 2.6.20: Pumping Rate. The time t required to empty a tank varies inversely ...
 2.6.21: Hookes Law. Hookes law states that the distance d that a spring wil...
 2.6.22: Relative Aperture. The relative aperture, or fstop, of a 23.5mm d...
 2.6.23: Musical Pitch. The pitch P of a musical tone varies inversely as it...
 2.6.24: Weight on Mars. The weight M of an object on Mars varies directly a...
 2.6.25: Find an equation of variation for the given situation.y varies inve...
 2.6.26: Find an equation of variation for the given situation.y varies inve...
 2.6.27: Find an equation of variation for the given situation.y varies dire...
 2.6.28: Find an equation of variation for the given situation.y varies dire...
 2.6.29: Find an equation of variation for the given situation.y varies join...
 2.6.30: Find an equation of variation for the given situation.y varies dire...
 2.6.31: Find an equation of variation for the given situation.y varies join...
 2.6.32: Find an equation of variation for the given situation.y varies join...
 2.6.33: Find an equation of variation for the given situation.y varies join...
 2.6.34: Find an equation of variation for the given situation.y varies join...
 2.6.35: Intensity of Light. The intensity I of light from a light bulb vari...
 2.6.36: Atmospheric Drag. Wind resistance, or atmospheric drag, tends to sl...
 2.6.37: Stopping Distance of a Car. The stopping distance d of a car after ...
 2.6.38: Weight of an Astronaut. The weight W of an object varies inversely ...
 2.6.39: EarnedRun Average. A pitchers earnedrun average E varies directly...
 2.6.40: Boyles Law. The volume V of a given mass of a gas varies directly a...
 2.6.41: In each of Exercises 4145, fill in the blank with the correct term....
 2.6.42: In each of Exercises 4145, fill in the blank with the correct term....
 2.6.43: In each of Exercises 4145, fill in the blank with the correct term....
 2.6.44: In each of Exercises 4145, fill in the blank with the correct term....
 2.6.45: In each of Exercises 4145, fill in the blank with the correct term....
 2.6.46: In each of the following equations, state whether y varies directly...
 2.6.47: Volume and Cost. An 18oz jar of peanut butter in the shape of a ri...
 2.6.48: Describe in words the variation given by the equation Q = kp2 q3 .
 2.6.49: Area of a Circle. The area of a circle varies directly as the squar...
Solutions for Chapter 2.6: Variation and Applications
Full solutions for College Algebra: Graphs and Models  5th Edition
ISBN: 9780321783950
Solutions for Chapter 2.6: Variation and Applications
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.6: Variation and Applications includes 49 full stepbystep solutions. College Algebra: Graphs and Models was written by and is associated to the ISBN: 9780321783950. This textbook survival guide was created for the textbook: College Algebra: Graphs and Models, edition: 5. Since 49 problems in chapter 2.6: Variation and Applications have been answered, more than 25865 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Iterative method.
A sequence of steps intended to approach the desired solution.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Solvable system Ax = b.
The right side b is in the column space of A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).